Geometry and Topology Seminar
The self-dual Yang-Mills-Higgs (or Ginzburg-Landau) functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly those satisfying a first-order system known as the "vortex equations" in the Kahler setting) have long been studied as a basic model problem in gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.